Finding the Pattern in the Number Series

The question asks us to find the next number in the given series: 21, 22, 18, 27, 11, 36, ?

To solve number series problems, we look for a pattern in the sequence of numbers. This pattern can involve addition, subtraction, multiplication, division, squares, cubes, or a combination of these operations between consecutive terms.

Let's examine the difference between consecutive terms:

• Difference between 22 and 21: \(22 - 21 = +1\)
• Difference between 18 and 22: \(18 - 22 = -4\)
• Difference between 27 and 18: \(27 - 18 = +9\)
• Difference between 11 and 27: \(11 - 27 = -16\)
• Difference between 36 and 11: \(36 - 11 = +25\)

The sequence of differences is +1, -4, +9, -16, +25. Let's look closer at the magnitudes of these differences: 1, 4, 9, 16, 25. These are perfect squares:

• \(1 = 1^2\)
• \(4 = 2^2\)
• \(9 = 3^2\)
• \(16 = 4^2\)
• \(25 = 5^2\)

Now let's look at the signs of the differences: +, -, +, -, +. The signs are alternating starting with a positive sign.

So, the pattern involves adding or subtracting consecutive perfect squares, starting with \(1^2\), and alternating the sign starting with plus.

The pattern of operations is:

• Term 1 to Term 2: Add \(1^2\) (\(21 + 1 = 22\))
• Term 2 to Term 3: Subtract \(2^2\) (\(22 - 4 = 18\))
• Term 3 to Term 4: Add \(3^2\) (\(18 + 9 = 27\))
• Term 4 to Term 5: Subtract \(4^2\) (\(27 - 16 = 11\))
• Term 5 to Term 6: Add \(5^2\) (\(11 + 25 = 36\))

Following this pattern, the next step (from Term 6 to Term 7) should involve subtracting the next perfect square, which is \(6^2\).

The next perfect square is \(6^2 = 36\).

The operation should be subtraction (as the signs alternate +, -, +, -, +, -). So we subtract 36 from the last term, which is 36.

Calculation for the next term:

\(36 - 6^2 = 36 - 36 = 0\)

Therefore, the next number in the series is 0.

Step-by-Step Solution

1. Identify the given series: 21, 22, 18, 27, 11, 36, ?

2. Calculate the differences between consecutive terms:

• \(22 - 21 = +1\)
• \(18 - 22 = -4\)
• \(27 - 18 = +9\)
• \(11 - 27 = -16\)
• \(36 - 11 = +25\)

3. Analyze the differences: +1, -4, +9, -16, +25.

4. Recognize the magnitudes as perfect squares: \(1=1^2\), \(4=2^2\), \(9=3^2\), \(16=4^2\), \(25=5^2\).

5. Observe the alternating signs: +, -, +, -, +.

6. Deduce the pattern: The series is formed by adding or subtracting successive perfect squares (\(1^2, 2^2, 3^2, \dots\)) with alternating signs starting with addition.

7. Determine the next operation: The next step is to subtract \(6^2\).

8. Calculate the next term: \(36 - 6^2 = 36 - 36 = 0\).

Series Pattern Summary

The number that replaces the question mark is 0.

Revision Table: Number Series Patterns

Additional Information: Solving Number Series Questions

Solving number series questions is a common type of problem in quantitative aptitude and logical reasoning tests. Here are some tips for approaching them:

• Look at the Differences: Calculate the differences between consecutive terms. This often reveals an arithmetic series, a series of squares/cubes, or another recognizable pattern.
• Look at the Ratios: If the numbers are increasing or decreasing rapidly, check the ratios between consecutive terms. This might indicate a geometric series or a pattern involving multiplication/division.
• Look for Alternating Patterns: Some series have two interleaved patterns or operations that alternate between terms.
• Consider Squares and Cubes: Be familiar with perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100...) and perfect cubes (1, 8, 27, 64, 125...). They frequently appear in series.
• Check for Prime Numbers: Sometimes the pattern involves prime numbers (2, 3, 5, 7, 11, 13...).
• Combine Operations: The pattern might involve a combination of operations, such as multiply by 2 and add 1, or subtract 3 and square the result.
• Break Down Complex Series: If the pattern isn't immediately obvious, try looking at the differences of the differences (second-order differences) or other transformations.
• Test Your Pattern: Once you identify a potential pattern, test it on the existing terms to ensure it holds true throughout the series before predicting the next term.

Practice with various types of number series problems will help you become quicker at identifying patterns.